(a) Show that f is one-to-one. (b) Use Theorem 7 to find (f^-1)^'(a). (c) Calculate f^-1(x) and

step 1 So our first step is to show that this function is one to one. We'll call our graph for x cubed.

(a) Show that f is one-to-one. (b) Use Theorem 7 to find...

(a) Show that $f$ is one-to-one. (b) Use Theorem 7 to find $\left(f^{-1}\right)^{\prime}(a)$. (c) Calculate $f^{-1}(x)$ and state the domain and range of $f^{-1}$. (d) Calculate $\left(f^{-1}\right)^{\prime}(a)$ from the formula in part (c) and check that it agrees with the result of part (b). (e) Sketch the graphs of $f$ and $f^{-1}$ on the same axes. $f(x)=x^3, \quad a=8$

(a) Show that $f$ is one-to-one.
(b) Use Theorem 7 to find $\left(f^{-1}\right)^{\prime}(a)$.
(c) Calculate $f^{-1}(x)$ and state the domain and range of $f^{-1}$.
(d) Calculate $\left(f^{-1}\right)^{\prime}(a)$ from the formula in part (c) and check that it agrees with the result of part (b).
(e) Sketch the graphs of $f$ and $f^{-1}$ on the same axes.

$f(x)=x^3, \quad a=8$

Key Concepts

Inverse Function Theorem

The Inverse Function Theorem provides the conditions under which a function has a differentiable inverse and gives a formula for the derivative of the inverse function. Specifically, if a function is differentiable and its derivative is non-zero at a point, then locally its inverse exists and the derivative of the inverse at the corresponding point is the reciprocal of the derivative of the original function.

One-to-One Function

A one-to-one function (injective function) is one in which every element of the function's domain maps to a unique element in the codomain. This property ensures that the function never takes the same value twice, which is a necessary condition for the function to have an inverse function.

Inverse Function

An inverse function reverses the mapping of the original function, meaning if f maps x to y, then the inverse function f?¹ maps y back to x. The concept of the inverse function relies on the original function being one-to-one, as this guarantees that each output is associated with a unique input.

Derivative of an Inverse Function

The derivative of an inverse function at a given point can be calculated using the formula (f?¹)'(y) = 1 / f'(x), where y = f(x) and f'(x) is the derivative of the original function. This calculation is significant for understanding how changes in the output of a function are reflected when the function is reversed.

Domain and Range

The domain of a function consists of all the values that can be input into the function, while the range consists of all the values that can be output. For an inverse function, the domain is the range of the original function and the range is the domain of the original function. This interchange is critical when analyzing the properties of the inverse function.

Graphical Interpretation of Inverse Functions

Graphing a function and its inverse on the same axes reveals that they are mirror images of each other with respect to the line y = x. This reflective symmetry provides a visual confirmation of the inverse relationship and helps in understanding how the function and its inverse interact.

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